Stability Regions and Bifurcations for Higher-Order Fractional Difference Equations

Abstract

We study stability regions for the higher-order, two-term fractional difference equation ${\Delta }^{\alpha }x(t)+a{\Delta }^{\beta }x(t+\alpha -\beta -1)=(b-1)x(t+\alpha -2)$, where $0<\beta \le 1<\alpha \le 2$, $a>0$, and $b\in \mathbb{C}$. The Z-transform yields a characteristic function whose image of the unit circle determines the stability boundary. Using a winding-number formulation, we give a necessary and sufficient root-count condition for asymptotic stability. Two analytically derived parameter values, $a_1=2^{\alpha -\beta }$ and $a_2=2^{\alpha -\beta }(4-\alpha )/(2-\beta )$, characterize an endpoint collision and a loss of regularity of the boundary curve, respectively. The real-parameter case and a nonlinear higher-order logistic map are treated as consequences of the same stability criterion. We also analyze the one-term family ${\Delta }^{\alpha }x(t)=(c-1)x(t+\alpha -N)$ for $N-1<\alpha \le N$. A winding-number bound proves that its stability region is empty for every $N\ge 3$. Numerical experiments illustrate the theoretical results.

23 pages, 43 figures